In this paper, a class of cosmological models able to describe the whole Universe expansion history as a succession of \(\Lambda CDM\)  expansions is found. Let us explain why we consider such a possibility. In a classical way, we assume that Universe is filled by matter, radiation and dark energy with respectively positive densities \(\rho _m\) , \(\rho _r\)  and \(\rho _d\)  (an assumption that could break down near the singularity). Defining their energy density parameters \(\Omega _r\) , \(\Omega _m\)  and \(\Omega _d\)  (see their definitions in the next section) such as \(\Omega _m+\Omega _r+\Omega _d=1\) , Universe evolution can then be represented as a trajectory in the space phase \((\Omega _m,\Omega _r)\) . From today to radiation dominated phases, our Universe is well described by a \(\Lambda CDM\)  model. Its trajectory in agreement with observations[1]} is shown in bold on the phase space of figure REF . One often describes its different epochs by the domination of radiation, matter and then dark energy. But its trajectory also encourages us to describe it as a two phases model, the first one with \(\Omega _d\simeq 0\)  and the second one with \(\Omega _r\simeq 0\) . Moreover, at early time, we should have an inflation phase with a very small matter density parameter \(\Omega _m\simeq 0\) . Inflation is not predicted by the \(\Lambda CDM\)  model but it could be due to a flat potential (effective, from a scalar field, etc), behaving like a cosmological constant with a value different from the one at our present time (solving the cosmological constant problem[2]}). An inflation phase could start with \(\Omega _d\simeq 1\)  and Universe should evolve to reach the radiation dominated phase \(\Omega _r\simeq 1\)  (solving the graceful exit problem[3]}). This inflation to radiation phase corresponds to the dashed trajectory on figure REF . Considering then this last trajectory associated to the bold \(\Lambda CDM\)  one, we observe that Universe evolution could thus be described as a close trajectory in the phase space \((\Omega _m,\Omega _r)\) , what is named an homoclinic orbit in dynamical system theory. Such a trajectory corresponds to some \(\Lambda CDM\)  expansions repeating periodically, separated by some inflation and radiation dominated phases (different from the standard \(\Lambda CDM\)  radiation phase). We call such a periodic behaviour of the expansion a "\(\Lambda CDM\)  periodic cosmology" and look for the families of cosmological models able to reproduce this scenario of the whole Universe history. Note that we choose to employ the word "periodic" instead of "cyclic" such that there is no confusion with the cyclic cosmological models[4]}, [5]}, [6]}, [7]} that avoid the beginning of time thanks to some bounces of the scale factor. This is generally not the case of a \(\Lambda CDM\)  periodic cosmology that has an initial singularity.
